Finite Group Databasebeta

← #10606622

Subgroups — #10606622

Order 640.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4083C211yesmaximal, characteristic
2C2×D4×Dic5C211yesmaximal, characteristic, complemented
3C22×#685C211yesmaximal, characteristic, complemented
4C2×#4032C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic, complemented
6#65501C211yesmaximal, characteristic, complemented
7#65250C211yesmaximal, characteristic, complemented
8#65250C211yesmaximal, characteristic, complemented
9#65247C211yesmaximal, characteristic, complemented
10#65244C211yesmaximal, characteristic, complemented
11#65239C211yesmaximal, characteristic, complemented
12#64966C211yesmaximal, characteristic, complemented
13#64963C211yesmaximal, characteristic, complemented
14#64959C211yesmaximal, characteristic, complemented
15#64959C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C23×Dic5C2211yescharacteristic, complemented
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic, complemented
25C2×C5×C22⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic, complemented
29#4083—24no—
30D4×Dic5—24no—
31C2×#685—24no—
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×C4⋊Dic5C2211yescharacteristic, complemented
43C2×C4⋊Dic5C2211yescharacteristic, complemented
44C2×C4⋊Dic5C2211yescharacteristic, complemented
45C2×C4⋊Dic5C2211yescharacteristic, complemented
46C2×C4⋊Dic5C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48#4032—24no—
49C5×#95C2211yescharacteristic, complemented
50C5×#95C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2418—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—22no—
63C23×D5—41no—
64C23×D5—41no—
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4D414yes—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—41no—
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C2×C22⋊D5—22no—
82C2×C22⋊D5—41no—
83C2×C22⋊D5—41no—
84C22×Dic5—22no—
85C22×Dic5—22no—
86C22×Dic5—22no—
87C22×Dic5—22no—
88C22×Dic5—41no—
89C22×Dic5—41no—
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
93C22×Dic5C2311yescharacteristic
94C22×Dic5C2311yescharacteristic
95C22×Dic5C2311yescharacteristic
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—41no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102#685—22no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—41no—
110#685—41no—
111#685D414yescomplemented
112C4⋊Dic5—22no—
113C4⋊Dic5—22no—
114C4⋊Dic5—22no—
115C4⋊Dic5—41no—
116C4⋊Dic5—41no—
117C4×Dic5—22no—
118C2×#123—51no—
119C2×#120—51no—
120C2×C4×D4—51no—
121C22×C22⋊C4—51no—
122#399—51no—
123#397—51no—
124#394—51no—
125#394—51no—
126#388—51no—
127#388—51no—
128#386—51no—
129#386D511yescharacteristic, complemented
130#385—51no—
131#380—51no—
132#379—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5C2×D412yes—
144C23×C5C2×D412yes—
145C23×C5#4112yes—
146C23×C5#4112yes—
147C23×C5C2411yescharacteristic
148C22×D5—28no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5—21no—
153C22×D5—21no—
154C22×D5—21no—
155C22×D5—21no—
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—21no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5—44no—
166C22×D5—44no—
167C22×D5C2×D412yes—
168C22×D5#4112yes—
169C5×D4—28no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—44no—
184C2×C4×C5C2×D412yes—
185C2×C4×C5C2×D412yes—
186C22⋊D5—44no—
187C22⋊D5—44no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—21no—
204C2×Dic5—21no—
205C2×Dic5—44no—
206C2×Dic5—44no—
207C2×Dic5C2×D412yes—
208C25—51no—
209C22×D4—51no—
210C22×D4—51no—
211C22×D4C2×D511yescharacteristic
212C23×C4—51no—
213C23×C4—51no—
214C23×C4C2×D511yescharacteristic
215#123—104no—
216#120—104no—
217C4×D4—104no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—51no—
222C2×C4⋊C4—51no—
223C2×C22⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—51no—
235C2×C22⋊C4—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic, complemented
239C2×C22⋊C4C2×D511yescharacteristic, complemented
240C2×C42—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95C2×D511yescharacteristic, complemented
246#95C2×D511yescharacteristic, complemented
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—22no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5#12314yescomplemented
261C22×C5C22×D411yescharacteristic
262C22×C5C22×D411yescharacteristic
263C22×C5C2×#4111yescharacteristic
264C22×C5C2×#4111yescharacteristic
265C22×C5C2×#4111yescharacteristic
266C22×C5H(5,𝔽2)11yescharacteristic
267C22×C5H(5,𝔽2)11yescharacteristic
268C2×D5—22no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—22no—
274C2×D5—416no—
275C2×D5—41no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—41no—
283C4×C5—22no—
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288Dic5—22no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—51no—
295C24—51no—
296C24—102no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—201no—
302C24—201no—
303C24C22×D511yescharacteristic
304C24C22×D511yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—201no—
312C2×D4—201no—
313C2×D4D2014yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—201no—
328C22×C4—201no—
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—102no—
335C4⋊C4—102no—
336C4⋊C4—102no—
337C4⋊C4—201no—
338C4⋊C4—201no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—41no—
342C22⋊C4—54no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—102no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C22⋊C4—102no—
353C22⋊C4—201no—
354C22⋊C4—201no—
355C42—102no—
356C2×C5—24no—
357C2×C5—42no—
358C2×C5C2×#12311yescharacteristic
359C2×C5#52511yescharacteristic
360C2×C5#53811yescharacteristic
361C2×C5D4211yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#55111yescharacteristic
365D5—42no—
366D5—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—52no—
378C23—52no—
379C23—108no—
380C23—101no—
381C23—101no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—204no—
393C23—204no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23C2×D2012yes—
399C23C2×D2012yes—
400C23#71112yes—
401C23#71112yes—
402C23C23×D511yescharacteristic
403D4—28no—
404D4—204no—
405D4—204no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—44no—
420C2×C4—52no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—204no—
439C2×C4—204no—
440C2×C4C2×D2012yes—
441C2×C4D4×D512yes—
442C5#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—2016no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22#403214yescomplemented
472C22C22×D2011yescharacteristic
473C22C2×D4×D511yescharacteristic
474C22C2×#71111yescharacteristic
475C22C2×#71111yescharacteristic
476C22#414411yescharacteristic
477C22D5×#4111yescharacteristic
478C22#414911yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—102no—
485C4—201no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—24no—
491C2—42no—
492C2—202no—
493C2—202no—
494C2C2×#403211yescharacteristic
495C2D4×D2011yescharacteristic
496C2#6588811yescharacteristic
497C2#6592511yescharacteristic
498C2#6594011yescharacteristic
499C2#6595111yescharacteristic
500C2#6599211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4083C2
hypercenter#386D5
hypocenterC5#2418
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×D4×Dic5C2semidirect
3. C22×#685C2semidirect
4. C2×#4032C2semidirect
5. C5×#386C2semidirect
6. #65501C2semidirect
7. #65250C2semidirect
8. #65250C2semidirect
9. #65247C2semidirect
10. #65244C2semidirect
11. #65239C2semidirect
12. #64966C2semidirect
13. #64963C2semidirect
14. #64959C2semidirect
15. #64959C2semidirect
22. C23×Dic5C22semidirect
24. C2×C5×C22⋊C4C22semidirect
25. C2×C5×C22⋊C4C22semidirect
28. C2×C22⋊Dic5C22semidirect
42. C2×C4⋊Dic5C22semidirect
43. C2×C4⋊Dic5C22semidirect
44. C2×C4⋊Dic5C22semidirect
45. C2×C4⋊Dic5C22semidirect
46. C2×C4⋊Dic5C22semidirect
47. C2×C4×Dic5C22semidirect
49. C5×#95C22semidirect
50. C5×#95C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
111. #685D4semidirect
129. #386D5semidirect
238. C2×C22⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
245. #95C2×D5semidirect
246. #95C2×D5semidirect
260. C22×C5#123semidirect
442. C5#2418semidirect
471. C22#4032semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#40832
C2×D4×Dic52
C22×#6852
C2×#40322
C5×#3862
#655012
#652502
#652472
#652442
#652392
#649662
#649632
#649592
#24185